{"paper":{"title":"The difference and ratio of the fractional matching number and the matching number of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ilkyoo Choi, Jaehoon Kim, Suil O","submitted_at":"2015-12-23T19:24:57Z","abstract_excerpt":"Given a graph $G$, the matching number of $G$, written $\\alpha'(G)$, is the maximum size of a matching in $G$, and the fractional matching number of $G$, written $\\alpha'_f(G)$, is the maximum size of a fractional matching of $G$. In this paper, we prove that if $G$ is an $n$-vertex connected graph that is neither $K_1$ nor $K_3$, then $\\alpha'_f(G)-\\alpha'(G) \\le \\frac{n-2}6$ and $\\frac{\\alpha'_f(G)}{\\alpha'(G)} \\le \\frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.07595","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}